Application of mathematics in sports

APPLICATION OF MATHEMATICS IN SPORTS

Mr. PUNDIKALA VEERESHA M Sc.

SPORTS

INTRODUCTION:

Sport (or sports) is all forms of usually

competitive physical activity which, through

casual or organized participation, aim to use

maintain or improve physical ability and

skills while providing entertainment to

participants.

Sports are usually governed by a set

of rules or customs, which serve to ensure

fair competition, and allow consistent

adjudication of the winner. Winning can be

determined by physical events such as

scoring goals or crossing a line first, or by

the determination of judges who are scoring

elements of the sporting performance.

HISTORY :

There are artifacts and structures that

suggest that the Chinese engaged in

sporting activities as early as 2000 BC.

Monuments to the Pharaohs indicate that a

number of sports, including swimming and

fishing, were well-developed and regulated

several thousands of years ago in ancient

Egypt.

A wide range of sports were already

established by the time of Ancient Greece

and the military culture and the

development of sports in Greece

influenced one another considerably.

Sports became such a prominent part of

their culture that the Greeks created the

Olympic Games,.

UNIT 2.1: ATHLETICS

INTRODUCTION AND HISTORY :

Athletics is an exclusive collection of sporting events that involve competitive

running, jumping, throwing, and walking. The most common types of athletics

competitions are track and field, road running, cross country running, and race walking.

The simplicity of the competitions, and the lack of a need for expensive equipment, makes

athletics one of the most commonly competed sports in the world.

Athletics events were depicted in the Ancient Egyptian tombs in Saqqara,

with illustrations of running at the HebSed festival and high jumping appearing in

tombs from as early as of 2250 BC. The Tailteann Games were an ancient Celtic

festival in Ireland, founded around 1800 BC, and the thirty-day meeting included

running and stone-throwing among its sporting events.

APPLICATIONS:

1. If a runner ran one lap of the track in the second lane, how far did she / he run?

Solution:

You might think that the answer is 400 meters, since it is a 400 meter track. Both straight-

aways are 100 meters but as you go further out from the inside rail on the semicircular turns, you

run farther. To calculate the distance around the turns, we use the formula for the circumference

(C) of a circle with radius (r): 𝐶 = 2𝜋𝑟 since the two semicircles on the inside rail form a circle

with a circumference of 200 m, we can find the radius of the inside rail semicircles.

𝐶 = 2𝜋𝑟 ⇒ 200 = 2𝜋𝑟

𝑟 ≈ 31.83

Now when you run in the second lane, the semicircle turns

have a radius that is one meter longer or 32.83 m Thus, distance

around both turns in the second lane is;

𝐶 = 2𝜋𝑟 = 2𝜋 32.83 ≈ 206.2 𝑚

.

Thus, a lap in the second lane has a total distance of

406.2 m, 200 meters for the straight-aways plus 206.2 m

for the turns.

2. A runner jogs ten laps of the track in the eighth lane. How far does he run?

a) Measured in meters. b) Measured in miles.

Solution:

a) Let us first determine the distance of one lap in the eighth lane. The straight-away are still 100

m each but the semicircular turns now have a radius that is 8 m more than the radius of the inside

rail.

𝑟 = 31.83 + 8 = 39.83 𝑚

The distance around both turns in the eighth lane is as follows.

𝐶 = 2𝜋𝑟 = 2𝜋(39.83) ≈ 250.1𝑚

Thus, a lap in the eighth lane is 450.1 m, 200 meters for the straight-aways plus 250.1 m for the

turns. Ten laps would be ten times that amount or 4501 m.

b) To change 4501 meters into miles we can set up a proportion comparing miles to meters using

the conversion, 6.2 mi ≈ 10,000 m.

Let x = the distance in miles

6.2 / 10,000 = 𝑥 / 4501 ⇒ 10,000 𝑥 = 27,906.2

𝑥 ≈ 2.8

By running ten laps in the eighth lane, the jogger ran about 2.8 miles.

3. An archery target consists of five concentric as shown . The value for an arrow in each region

starting from the inner circle is 9, 7, 5, 3, 1 points. In how many ways could five scoring arrows

exam 29 points?

Solution:

We can set up Diophantine equations to model the problem.

Let a = the number of 9 point arrows

b = the number of 7 point arrows

c = the number of 5 point arrows

d = the number of 3 point arrows

e = the number of 1 point arrows. Since there are five scoring arrows,

𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 = 5.

Since the five arrows score 29 points,

9𝑎 + 7𝑏 + 5𝑐 + 3𝑑 + 1 𝑒 = 29

We can now set up a table of values and systematically

find all possible values that simultaneously satisfy both equations.

We can start with the largest value for ‘a’ and determine the

values for the other variables making sure we use whole numbers and

a total of five arrows.

a(9) b(7) c(5) d(3) e (1)

3 0 0 0 2

2 0 2 0 1

2 1 0 1 1

1 2 1 0 1

1 2 0 2 0

1 1 2 1 0

1 0 4 0 0

0 4 0 0 1

0 3 1 1 0

0 2 3 0 0

There are ten ways to attain a score of 29 with five arrows.

The examples in this section have shown you that math can be used in analyzing sport stats and scoring.

The maximum value for a is 3, since when ‘a’ reaches 4, 9𝑎 = 36, which is over the 29

total points.

UNIT 2.2: BASKET BALL

INTRODUCTION AND HISTORY :

Basketball is a sport played by two teams of five players on a rectangular court. The

objective is to shoot a ball through a hoop 18 𝑖𝑛𝑐ℎ𝑒𝑠 (46 𝑐𝑚) in diameter and

10 𝑓𝑒𝑒𝑡 (3.0 𝑚) high mounted to a backboard at each end. Basketball is one of the world's

most popular and widely viewed sports.

In early December 1891, Canadian American Dr. James Naismith, a physical

education professor and instructor at the International Young Men's Christian Association

Training School (YMCA) (today, Springfield College) in Springfield, Massachusetts, (USA).

In 1922, the Commonwealth Five, the first all-black professional team was founded.

The New York Renaissance was founded in 1923.In 1939; the all-black New York

Renaissance beat the all-white Oshkosh All-Stars in the World Pro Basketball Tournament.

APPLICATIONS:

We know the equation

𝑓 𝑥 =−16

𝑣02 𝑐𝑜𝑠2𝛼

𝑥2 + 𝑡𝑎𝑛𝛼 𝑥 + ℎ0

Where

ℎ0 = Height from which the ball is thrown, 𝛼 = Angle at which the ball is thrown,

𝑣0 = Speed at which the ball is thrown, 𝑥 = Distance that the ball travels,

To help figure out the velocity at which a basketball player must throw the ball in order for it to

land perfectly in the basket. When shooting a basketball you want the ball to hit the basket at as

close to a right angle as possible. For this reason, most players attempt to shoot the ball at

a 45°angle. To find the velocity at which a player would need to throw the ball in order to make

the basket we would want to find the range of the ball when it is thrown at a 45° angle.

The formula for the range of the ball is

𝑅𝑎𝑛𝑔𝑒 =𝑣02sin(2𝛼)

32

But since for angle at which the ball is thrown is 𝛼 = 45° (angle), we have

𝑅𝑎𝑛𝑔𝑒 =𝑣02sin(2𝛼)

32=𝑣02sin(2 × 45)

32=

𝑣02

32

Now, if a player is shooting a 3 point shot, then he is approximately 25 𝑓𝑒𝑒𝑡 from the basket. If

we look at the graph of the range function we can get an idea of how hard the player must throw

the ball in order to make a 3 point.

So, by solving the formula knowing that the range of the shot must be 25𝑓𝑒𝑒𝑡,

25 =𝑣02

32⇒ 𝑣0

2= 800

𝑣0 ≈ 28.2843

So in order to make the 3 point shot, the player must throw the ball at approximately

28 𝑓𝑒𝑒𝑡 per second,

1. A basketball player realizes that in order to be more competitive in salary negotiations, he needs

to complete at least 88% of his free throw attempts by the end of the season. After 50 games, he

has made 173 out of 202 attempted free throws. If in the last 42 games of the season, he expects to

have 168 more free throws, how many of those does he have to make in order to reach the 88% free

throw mark?

Solution:

Let 𝑥 = the number of free throws he needs to make out of 168.

173 + 𝑥 = the number of free throws made

202 + 168 = the number of free throws attempted

173 + 𝑥

370= ratio of free throws made to attempted

To find x set that ratio equal to 88% = 0.88 and solve.

173 + 𝑥

370= 0.88

173 + 𝑥 = 325.6

⇒ 𝑥 = 152.6 ≈ 153

Thus, by making 153 out of his last 168 free throws, the player will reach the 88% mark.

UNIT 2.3: BASEBALL

INTRODUCTION AND HISTORY :

Baseball is a bat-and-ball game played between two teams of nine players who take

turns batting and fielding. The offense attempts to score more runs than its opponents by hitting a

ball thrown by the pitcher with a bat and moving counter-clockwise around a series of four bases:

first, second, third and home plate. A run is scored when the runner advances around the bases

and returns to home plate.

David Block discovered that the first recorded game of “Bass-Ball” took place in 1749

in Surrey, and featured the Prince of Wales as a player. William Bray, an English lawyer,

recorded a game of baseball on Easter Monday 1755 in Guildford, Surrey.

By the early 1830s, there were reports of a variety of un-codified bat-and-ball games

recognizable as early forms of baseball being played around North America.

APPLICATIONS:

1. A professional baseball team has won 62 and lost 70 games. How many consecutive wins

would bring them to the 500 mark?

Solution:

Let 𝑥 = The number of consecutive wins.

62 + 𝑥 = The number of wins

62 + 70 + 𝑥 = The total games played

( 62 + 𝑥 ) / (132 + 𝑥) = Ratio of wins to total games

Since the “500” mark is 50%, set the ratio equal to 0.50and solve for x

( 62 + 𝑥 ) / ( 132 + 𝑥 ) = 0.50

62 + 𝑥 = 66 + 0.50 𝑥

0.5 𝑥 = 4

𝑥 = 8

By winning 8 consecutive games, the team would reach the “500” mark

The Pythagorean Theorem:

There are many instances when distances in sports can be

determined using the Pythagorean Theorem.

This theorem states for a right triangle with legs a and b and hypotenuse c,

𝑎2 + 𝑏2 = 𝑐2

2. How long is the throw from third base to first base on a professional baseball diamond where

the bases are 90 feet apart?

Solution:

A right triangle is formed by the third base line and the first base line with the throw

from third to first is the hypotenuse of the right triangle, if d represents the distance from third

base to first base, using the Pythagorean Theorem, we get the following;

𝑎2 + 𝑏2 = 𝑑2

902 + 902 = 𝑑2 ⇒ 16200 = 𝑑2

127.3 ≈ 𝑑

UNIT 2.4: GOLF

The Mathematics of Golf

INTRODUCTION AND HISTORY :

Golf is a sport that many people enjoy and there are more math aspects of the game

than most people can imagine. The winner was whoever hit the ball with the least number of

strokes into a target several hundred yards away.

A golf-like game is, apocryphally, recorded as taking place on 26 February 1297, in

Leonean de-Vecht, where the Dutch played a game with a stick and leather ball was also played

in 17th-century Netherlands and that this predates the game in Scotland. There are also other

reports of earlier accounts of a golf-like game from continental Europe.

APPLICATIONS:

1. Professional golfers frequently drive a golf ball 300 or more yards, but very few have an

average distance of 300 or more yards for the year. For example, at one point in a season,

Tiger woods had an average of 293.4 𝑦𝑎𝑟𝑑𝑠 out of 104 officially measured drives. What

does he have to average on his next 40 measured drives to bring his average for the yard up

to 300 𝑦𝑒𝑎𝑟𝑠?

Solution:

To find the average distance, find the total yards of all his drives divided by the number of drives.

Let 𝑥 = the average of the next 40 drives.

Total yards = 104 drives at 293.4 yards plus 40 drives at x yards

= 104(293.4) + 40𝑥 = 30,513.6 + 40𝑥

Number of drives = 104 + 40 = 144

Average = total yards/number of drives

= (30,513.6 + 40𝑥)/144

Setting the average to 300 and solving for x, we get:

(30,513.6 + 40𝑥)/144 = 300 ⇒ 30,513.6 + 40𝑥 = 43,200

𝑥 = 317.16

Thus, Tiger Woods must average 317.16 yards on his next 40 measured drives to average 300 yards for year.

2. Place-kicker scored a school record of 17 points with field goals (3 points each) and extra

points (1 point each). How many different ways could he have scored the 17 points ?

Solution:

We could use a guessing process and determine the different ways to score 17 points, for

example, 4 field goals and 5 extra points or 5 field goals and 2 extra points. However, if we are not

careful in our analysis, we might miss some of the possible solutions.

Let 𝑥 = the number of field goals scored.

𝑦 = the number of extra points scored.

Since each field goal is 3 points and each extra point is one point, we get the following.

3𝑥 + 1𝑦 = 17

There are some logical restrictions on x and y: both must be whole numbers;

The maximum value for 𝑥 = 5 since 6 field goals would be 18 points; and the maximum value for

𝑦 = 17.

Thus, the solution to the problem becomes solving the equation

3𝑥 + 1𝑦 = 17 where x and y are whole numbers ( 0 ≤ 𝑥 ≤ 5, 𝑎𝑛𝑑 0 ≤ 𝑦 ≤ 17)

Consider a table for x and y and systematically find all solutions by assigning an integer for x from 0 to

5 and calculating the value for y.

X (3 points) Y ( 1 points)

0 17

1 14

2 11

3 8

4 5

5 2

There are six ways in which the kicker could score 17 points.

Such an equation with integer coefficients and more than one integer solution is called a

Diophantine Equation after the Greek mathematician, Diophantus (c. 250 A.D).

UNIT 2.5: FOOTBALL

INTRODUCTION AND HISTORY :

Football refers to a number of sports that involve, to varying degrees, kicking a ball

with the foot to score a goal. The most popular of these sports worldwide is association football,

more commonly known as just “football” or “soccer”.

The Ancient Greeks and Romans are known to have played many ball games, some of

which involved the use of the feet. The Roman game harpastum is believed to have been adapted

from a Greek team game known as “ἐπίσκυρος” (Episkyros), which is mentioned by a Greek

playwright, Antiphanes (388–311 BC). These games appear to have resembled rugby football.

The Roman politician Cicero (106–43 BC) describes the case of a man who was killed whilst

having a shave when a ball was kicked into a barber's shop.

APPLICATIONS:

1. If a college football player kicks a field-goal when the ball was put in play 40 yards from the

goal line, what is the maximum length of a field goal as measured along the ground?

Solution:

To find the maximum length, we have to take the following into consideration;

1. Field goals are kicked from a point 7 yards behind the scrimmage line.

2. The field goal is kicked from the right hash mark and cleared the left goal post.

3. From the diagram of the football field below, we see that:

• The goal is 10 yards deep in the end zone

• The hash mark is 60 ft. from the side line

• The left goal post is 89.25 ft. from the side line.

Let 𝑥 = the distance to the farthest goal post along the ground.

The ball was kicked 7 yards behind the scrimmage line and there

are 40 yards to the goal line and 10 yards more to the end line where

the goal post is located. Thus, the perpendicular distance from the point

which the ball was kicked to the end line is 7 + 40 + 10 = 57 yards or

171 feet. The distance along the end line from the left goal post to the

perpendicular is 89.25 ft. – 60 ft. = 29.25 ft. The right triangle

formed can be solved using Pythagorean theorem.

𝑥2= 29.252 + 1712

𝑥2 = 30096.5625

𝑥 ≈ 173.5 𝑓𝑡 ≈ 57.8 𝑦𝑑

Thus, even though the ball was kicked with a 40-yard scrimmage line, the maximum distance

of the kick as measured along the ground is about 57.8 yards.

2. A football player runs 40 yards in 4.2 seconds, what is his speed

(a) In feet per second and

(b) Miles per hour?

Solution:

(a) We are given the distance in yards but we need the distance in feet to represent the rate in feet

per second. To convert yards to feet we can set up and solve a proportion comparing yards to feet.

Using the conversion fact,1 𝑦𝑑 = 3 𝑓𝑡, we get:

(d = the distance in feet)

1 𝑦𝑑/ 3 𝑓𝑡 = 40 𝑦𝑑 / 𝑑 𝑓𝑡

1/3 = 40 / 𝑑

1𝑑 = 3 (40)

𝑑 = 120 𝑓𝑡

Thus, the football player ran a distance of 120 𝑓𝑡 (𝑑 = 120) in 4.2 seconds (t = 4.2)

and the rate (r) is 𝑟 = 𝑑 / 𝑡 = 120 𝑓𝑡 / 4.2 ≈ 28.6 𝑓𝑡 /𝑠𝑒𝑐

(b) To find the rate in miles per hour, we can change the distance, 120 𝑓𝑡, into miles and the

time, 4.4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, into hours. As before, we do this by setting up and solving proportions using

the conversion facts, 1 𝑚𝑖 = 5280 ft and 1 ℎ𝑟 = 3600 𝑠𝑒𝑐.

Distance:

Let 𝑑 = the distance in miles

1 𝑚𝑖𝑙𝑒𝑠 / 5280 𝑓𝑡 = 𝑑 𝑚𝑖𝑙𝑒𝑠 / 120 𝑓𝑡

1 / 5280 = 𝑑 / 120

𝑑 = 0.022727 𝑚𝑖

Time: Let 𝑡 = the time in hours

1 ℎ𝑟 / 3600 𝑠𝑒𝑐 = 𝑡 ℎ𝑟 / 4.4 𝑠𝑒𝑐

1 / 3600 = 𝑡 / 4.4

𝑡 ≈ 0.001222 ℎ𝑟

The rate (r) is 𝑟 = 𝑑 / 𝑡 = 0.022727 𝑚𝑖 / 0.001222 ℎ𝑟

𝑟 ≈ 18.6 𝑚𝑝ℎ